Open Mapping and Closed Mapping

Open Mapping:

A mapping f from one topological space X into another topological space Y is said to be an open mapping if for every open set O in X, f\left( O \right) is open in Y. In other words a mapping is open if it carries open sets over to open sets. A continuous mapping may not be open.

Example: Let X = \left\{ {x,y,z} \right\} with topology {\tau _X} = \left\{ {\phi ,X,\left\{ y \right\},\left\{ {x,y} \right\},\left\{ {y,z} \right\}} \right\} and let Y = \left\{ {1,2,3} \right\} with topology {\tau _Y} = \left\{ {\phi ,Y,\left\{ 1 \right\}} \right\}, then f:X \to Y defined by f\left( x \right) = 2, f\left( y \right) = 1, f\left( z \right) = 3 is continuous but not open, because A = \left\{ {x,y} \right\} is open in X but f\left( A \right) = \left\{ {1,2} \right\} is not open in Y.

Closed Mapping:

A mapping f from one topological space X into another topological space Y is said to be an closed mapping if for every closed set G in X, f\left( G \right) is closed in Y. In other words a mapping is closed if it carries closed sets over to closed sets.

Theorems:
• Let X and Y be topological spaces. A bijective mapping f:X \to Y is open if and only if {f^{ - 1}}:Y \to X is continuous.
• Let X and Y be topological spaces. A bijective mapping f:X \to Y is closed if and only if {f^{ - 1}}:Y \to X is continuous.
• Let X and Y be topological spaces. A mapping f:X \to Y is open if and only if {\left[ {f\left( A \right)} \right]^o} = f\left( A \right) for every open subset A of X.